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Related to enumeration of alkane systems - see reference for precise definition.
2

%I #40 May 14 2024 00:45:44

%S 1,1,1,4,19,91,476,2586,14421,82225,476913,2804880,16689036,100276894,

%T 607588840,3708251888,22776251835,140676848445,873210347555,

%U 5444307431052,34080036632565,214104150405915,1349504948018208,8531467913710560,54083412667272300,343715994386622918,2189505804590364876

%N Related to enumeration of alkane systems - see reference for precise definition.

%C a(n) is the "number of all staggered conformers of alkyls containing n carbon atoms". It is related to sequence b(n) = A001764(n), which is the number of "space positions of conformers of alkyls related to another alkyl without C_3 symmetry" that contain n carbon atoms. The generating functions of the sequences (a(n): n >= 0) and (b(n): n >= 0), with a(0) = b(0) = 1, appear in some of the papers below. - _Petros Hadjicostas_, Jul 24 2019

%H S. J. Cyvin, <a href="https://doi.org/10.1007/BF01164853">Algebraic solution for the numbers of staggered conformers of alkanes</a>, J. Math. Chem. 17 (1995), 291-293.

%H S. J. Cyvin, J. Brunvoll, B. N. Cyvin, and W. Lüttke, <a href="https://doi.org/10.1515/zna-1995-0911">Enumeration of the staggered conformers of alkanes</a>, Zeitschrift für Naturforschung A 50(9) (1995), 857-863.

%H S. J. Cyvin, B. N. Cyvin, J. Brumvoll, and Jianji Wang, <a href="https://doi.org/10.1016/S0022-2860(97)00419-5">Enumeration of staggered conformers of alkanes and monocyclic cycloalkanes</a>, J. Molec. Struct. 445 (1998), 127-137.

%H S. J. Cyvin, Jianji Wang, J. Brunvoll, Shiming Cao, Ying Li, B. N. Cyvin, and Yugang Wang, <a href="https://doi.org/10.1016/S0022-2860(97)00025-2">Staggered conformers of alkanes: complete solution of the enumeration problem</a>, J. Molec. Struct. 413-414 (1997), 227-239.

%H F. Hering et al., <a href="http://dx.doi.org/10.1016/0012-365X(82)90121-2">The enumeration of stack polytopes and simplicial clusters</a>, Discrete Math., 40 (1982), 203-217.

%H Jianji Wang, Shiming Cao, and Ying Li, <a href="https://doi.org/10.1007/BF01165165">An algebraic solution for the numbers of staggered conformers of alkanes</a>, J. Math. Chem. 20 (1996), 211-212.

%F From _Petros Hadjicostas_, Jul 24 2019: (Start)

%F We have a(0) = 1, while for n >= 1 we have

%F a(n) = (1/3) * A001764(n) = binomial(3*n, n)/(3*(2*n + 1)) if n !== 1 (mod 3), and

%F a(n) = (1/3) * A001764(n) + (2/3) * A001764((n-1)/3) if n == 1 (mod 3).

%F G.f.: 1 + (x/3) * (B(x)^3 + 2*B(x^3)), where B(x) is the g.f. of sequence A001764, which satisfies the functional equation B(x) = 1 + x*B^3(x). (It also satisfies the equation B(x) = 1/(1 - x*B^2(x)).) We have

%F B(x) = (2/sqrt(3*x)) * sin((1/3) * arcsin(sqrt(27*x/4))).

%F (End)

%F The g.f. for this sequence is F(3,t) on page 209 of the Hering link. - _Robert A. Russell_, May 12 2024

%p A121179 := proc(n)

%p if n = 0 then

%p return 1;

%p elif modp(n,3) <> 1 then

%p A001764(n) ;

%p else

%p A001764(n)+2*A001764((n-1)/3) ;

%p end if;

%p %/3 ;

%p end proc:

%p seq(A121179(n),n=0..30) ; # _R. J. Mathar_, Jul 31 2019

%t b[n_] := Binomial[3n, n]/(2n + 1);

%t a[n_] := If[n == 0, 1, If[Mod[n, 3] != 1, b[n], b[n] + 2 b[(n-1)/3]]/3];

%t a /@ Range[0, 30] (* _Jean-François Alcover_, Apr 01 2020 *)

%Y Cf. A001764, A121190.

%K nonn

%O 0,4

%A _N. J. A. Sloane_, Aug 15 2006

%E More terms from _Petros Hadjicostas_, Jul 24 2019